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led to the remark, that on those nights in which the discordance, was greatest a strong wind' was blowing nearly from one station to the other. Allowing for the obvious effect of this, or rather eliminating it altogether, the mean velocities on different evenings were found to agree very closely.

332. It may perhaps be advisable to say a few words here about the use of hypotheses, and especially those of very different gradations of value which are promulgated in the form of Mathematical Theories of different branches of Natural Philosophy.

333. Where, as in the case of the planetary motions and disturbances, the forces concerned are thoroughly known, the mathematical theory is absolutely true, and requires only analysis to work out its remotest details. It is thus, in general, far ahead of observation, and is competent to predict effects not yet even observed—as, for instance, Lunar Inequalities due to the action of Venus upon the Earth, etc. etc., to which no amount of observation, unaided by theory, would ever have enabled us to assign the true cause. It may also, in such subjects as Geometrical Optics, be carried to developments far beyond the reach of experiment; but in this science the assumed bases of the theory are only approximate, and it fails to explain in all their peculiarities even such comparatively simple phenomena as Halos and Rainbows; though it is perfectly successful for the practical purposes of the maker of microscopes and telescopes, and has, in these cases, carried the construction of instruments to a degree of perfection which merely tentative processes never could have reached.

334. Another class of mathematical theories, based to a certain extent on experiment, is at present useful, and has even in certain cases pointed to new and important results, which experiment has subsequently verified. Such are the Dynamical Theory of Heat, the Undulatory Theory of Light, etc. etc. In the former, which is based upon the experimental fact that heat is motion, many formulae are at present obscure and uninterpretable, because we do not know what is moving or how it moves. Results of the theory in which these are not involved, are of course experimentally verified. The same difficulties exist in the Theory of Light. But before this obscurity can be perfectly cleared up, we must know something of the ultimate, or molecular, constitution of the bodies, or groups of molecules, at present known to us only in the aggregate.

335. A third class is well represented by the Mathematical Theories of Heat (Conduction), Electricity (Statical), and Magnetism (Permanent). Although we do not know how Heat is propagated in bodies, nor what Statical Electricity of Permanent Magnetism are, the laws of their forces are as certainly known as that of Gravitation, and can therefore like it be developed to their consequences, by the application of Mathematical Analysis. The works of Fourier',

· Théorie Analytique de la Chaleur. Paris, 1822

Green', and Poisson', are remarkable instances of such develop ment. Another good example is Ampère's Theory of Electrodynamics.

336. Mathematical theories of physical forces are, in general, of one of two species. First, those in which the fundamental assumption is far more general than is necessary. Thus the celebrated equation of Laplace's Functions contains the mathematical foundation of the theories of Gravitation, Statical Electricity,-Permanent Magnetism, Permanent Flux of Heat, Motion of Incompressible Fluids, etc. etc., and has therefore to be accompanied by limiting considerations when applied to any one of these subjects.

337. Again, there are those which are built upon a few experiments, or simple but inexact hypotheses, only; and which require to be modified in the way of extension rather than limitation. As a notable example, we may refer to the whole subject of Abstract Dynamics, which requires extensive modifications (explained in Division III.) before it can, in general, be applied to practical purposes.

338. When the most probable result is required from a number of observations of the same quantity which do not exactly agree, we must appeal to the mathematical theory of probabilities to guide us to a method of combining the results of experience, so as to eliminate from them, as far as possible, the inaccuracies of observation. But it must be explained that we do not at present class as inaccuracies Of observation any errors which may affect alike every one of a series of observations, such as the inexact determination of a zero-point or of the essential units of time and space, the personal equation of the observer, etc. The process, whatever it may be, which is to be employed in the elimination of errors, is applicable even to these, but only when several distinct series of observations have been made, with a change of instrument, or of observer, or of both.

339. We understand as inaccuracies of observation the whole class of errors which are as likely to lie in one direction as another in successive trials, and which we may fairly presume would, on the average of an infinite number of repetitions, exactly balance each other in excess and defect. Moreover, we consider only errors of such a kind that their probability is the less the greater they are; so that such errors as an accidental reading of a wrong number of whole degrees on a divided circle (which, by the way, can in general be probably corrected by comparison with other observations) are not to be included.

340. Mathematically considered, the subject is by no means an easy one, and many high authorities have asserted that the reasoning employed by Laplace, Gauss, and others, is not well founded ; although the results of their analysis have been generally accepted. As an excellent treatise on the subject has recently been published by Airy,

Essay on the Application of Math-matical Analysis to the Theories of Electricity and Magnetism. Nottingham, 1828. Reprinted in Crella's Journal.

* Mémoires sur le Magnetisme. Mém. de l'Acad. des Sciences, 1815.

it is not necessary for us to do more than sketch in the most cursory manner what is called the Method of Least Squares.

341. Supposing the zero-point and the graduation of an instrument (micrometer, mural circle, thermometer, electrometer, galvanometer, etc.) to be absolutely accurate, successive readings of the value of a quantity (linear distance, altitude of a star, temperature, potential, strength of an electric current, etc.) may, and in general do, continually differ. What is most probably the true value of the observed quantity ?

The most probable value, in all such cases, if the observations are all equally reliable, will evidently be the simple mean; or if they are not equally reliable, the mean found by attributing weights to the several observations in proportion to their presumed exactness. But if several such means have been taken, or several single observations, and if these several means or observations have been differently qualified for the determination of the sought quantity (some of them being likely to give a more exact value than others), we must assign theoret. ically the best method of combining them in practice.

342. Inaccuracies of observation are, in general, as likely to be in excess as in defect. They are also (as before observed) more likely to be small than great; and (practically) large errors are not to be expected at all, as such would come under the class of avoidable mistakes. It follows that in any one of a series of observations of the same quantity the probability of an error of magnitude x, must depend upon x*, and must be expressed by some function whose value diminishes very rapidly as x increases. The probability that the error lies between x and x + 8x, where ex is ver; small, must also be proportional to dx. The law of error thus found is

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NA h where h is a constant, indicating the degree of coarseness or delicacy of the system of measurement employed. The co-efficient to secures that the sum of the probabilities of all possible errors shall be unity, as it ought to be.

343. The Probable Error of an observation is a numerical quantity such that the error of the observation is as likely to exceed as to fall short of it in magnitude.

If we assume the law of error just found, and call P the probable error in one trial, we have the approximate result

P=0'477 h. 344. The probable error of any given multiple of the value of an observed quantity is evidently the same multiple of the probable error of the quantity itself.

The probable error of the sum or difference of two quantities, affected by independent errors, is the square root of the sum of the squares of their separate probable errors.

345. As above remarked, the principal use of this theory is in the deduction, from a large series of observations, of the values of the quantities sought in such a form as to be liable to the smallest pro bable error. As an instance-by the principles of physical astronomy, the place of a planet is calculated from assumed values of the elements of its orbit, and tabulated in the Nautical Almanac. The observed places do not exactly agree with the predicted places, for two reasons -first, the data for calculation are not exact (and in fact the main object of the observation is to correct their assumed values); second, the observation is in error to some unknown amount. Now the difference between the observed, and the calculated, places depends on the errors of assumed elements and of observation. Our methods are applied to eliminate as far as possible the second of these, and the resulting equations give the required corrections of the elements.

Thus if ö be the calculated R.A. of a planet: da, de, dw, etc., the corrections required for the assumed elements: the true R.A. is

0 + Ada + Ede + ido + etc., where A, E, II, etc., are approximately known. Suppose the observed R.A. to be ©, then

A + Ada + Ede + II dw + = ,

Aδα + Εδε + Πδα +... 0-0, a known quantity, subject to error of observation. Every observation made gives us an equation of the same form as this, and in general the number of observations greatly exceeds that of the quantities da, de, dw, etc., to be found.


346. The theorems of $ 344 lead to the following rule for com. bining any number of such equations which contain a smaller number of unknown quantities :

Make the probable error of the second member the same in each equution, by the employment of a proper factor: multiply each equation by the co-efficient of x in it and add all, for one of the final equations; and so, with reference to ý, %, etc., for the others. The probable errors of the Values of x, y, etc., found from these final equations will be less than those of the values derived from any other linear method of combining the equations.

This process has been called the method of Least Squares, because the values of the unknown quantities found by it are such as to render the sum of the squares of the errors of the original equations a minimum.

347. When a series of observations of the same quantity has been made at different times, or under different circumstances, the law connecting the value of the quantity with the time, or some other variable, may be derived from the results in several waysmall more or less approximate. Two of these methods, however, are so much more extensively used than the others, that we shall devote a page or

two here to a preliminary notice of them, leaving detailed instances of their application till we come to Heat, Electricity, etc. They consist in (i) a Curve, giving a graphic representation of the relation between the ordinate and abscissa, and (2) an Empirical Formula connecting the variables.

348. Thus if the abscissae represent intervals of time, and the ordinates the corresponding height of the barometer, we may con. struct curves which show at a glance the dependence of barometric; pressure upon the time of day; and so on. Such curves may be accurately drawn by photographic processes on a sheet of sensitive paper placed behind the mercurial column, and made to move past it with a uniform horizontal velocity by clockwork. A similar process is applied to the Temperature and Électricity of the atmosphere, ‘and to the components of terrestrial magnetism.

349. When the observations are not, as in the last section, continuous, they give us only a series of points in the curve, from which, however, we may in general approximate very closely to the result of continuous observation by drawing, liberâ manu, a curve passing through these points. This process, however, must be employed with great caution ; because, unless the observations are sufficiently close to each other, most important fluctuations in the curve may escape notice. It is applicable, with abundant accuracy, to all cases where the quantity observed changes very slowly. Thus, for instance, weekly observations of the temperature at depths of from 6 to 24 feet underground were found by Forbes sufficient for a very accurate approximation to the law of the phenomenon.

350. As an instance of the processes employed for obtaining an empirical formula, we may mention methods of Interpolation, to which the problem can always be reduced. Thus from sextant observations, at known intervals, of the altitude of the sun, it is a common problem of Astronomy to determine at what instant the altitude is greatest, and what is that greatest altitude. The first enables us to find the true solar time at the place, and the second, by the help of the Nautical Almanac, gives the latitude. The calculus of finite differences gives us formulae proper for various data; and Lagrange has shown how to obtain a very useful one by elementary algebra., In finite differences we have

f(x + h) =f(x) + haf (x) +1;") of (*) +... This is useful, inasmuch as the successive differences, A/(x), A’f (w), etc., are easily calculated from the tabulated results of observation, provided these have been taken for equal successive in. crements of x.

If for values x, Xy... Xn, a function takes the values y, Yoga Yg,... Yn, Lagrange gives for it the obvious expression

+ ...) (20 – ) (25 – xg)...(ze 614 -2g) (2 )... (27 – Un) 2 - xq (tg - X;) (22 - 2.2) ... {*, - xn)


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